We study a class of two-stage stochastic integer programs with general integer variables in both stages and finitely many realizations of the uncertain parameters. Based on Benders' method, we propose a decomposition algorithm that utilizes Gomory cuts in both stages. The Gomory cuts for the second-stage scenario subproblems are parameterized by the first-stage decision variables, i.e., they are valid for any feasible first-stage solutions. In addition, we propose an alternative implementation that incorporates Benders' decomposition into a branch-and-cut process in the first stage. We prove the finite convergence of the proposed algorithms. We also report our preliminary computations with a rudimentary implementation of our algorithms to illustrate their effectiveness.

Abstract. In this paper, we study a multi-echelon uncapacitated lot-sizing problem in series (m-ULS), where the output of the intermediate echelons has its own external demand, and is also an input to the next echelon. We propose a polynomial-time dynamic programming algorithm, which gives a tight, compact extended formulation for the two echelon case (2-ULS). Next, we present a family of valid inequalities for m-ULS, show its strength and give a polynomial-time separation algorithm. We establish a hierarchy between the alternative formulations for 2-ULS. In particular, we show that our valid inequalities can be obtained from the projection of the multi-commodity formulation. Our computational results show that this extended formulation is very effective in solving our uncapacitated multi-item 2-echelon test problems. In addition, for capacitated multiitem multi-echelon problems, we demonstrate the effectiveness of a branch-and-cut algorithm using the proposed inequalities.

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In this paper, we consider a finite-horizon stochastic mixed-integer program involving dynamic decisions under a constraint on the overall performance or reliability of the system. We formulate this problem as a multi-stage (dynamic) chance-constrained program, whose deterministic equivalent is a large-scale mixedinteger program. We study the structure of the formulation, and develop a branch-and-cut method for its solution. We illustrate the efficacy of the proposed model and method on a dynamic inventory control problem with stochastic demand in which a specific service level must be met over the entire planning horizon. We compare our dynamic model with a static chance-constrained model, a dynamic risk-averse optimization model, a robust optimization model, and a pseudo-dynamic approach, and show that significant cost savings can be achieved at high service levels using our model.

In this paper, we consider physician scheduling problems originating from a medical staff scheduling service provider based in the United States. Creating a physician schedule is a complex task. An optimal schedule must balance a number of goals including adequately staffing required assignments for quality patient care, adhering to a unique set of rules that depend on hospital and medical specialties, and maintaining a work-life balance for physicians. We study various types of physician and hospital requirements with different priorities, including equalization constraints to ensure that each provider will receive approximately the same number of a specified shift over a given time period. A major challenge involves ensuring an equal distribution of workload among physicians, with the end goal of producing a schedule that will be perceived by physicians as fair while still meeting all other requirements for the group. As the number of such equalization constraints increases, the physician scheduling optimization problem becomes more complex and it requires more time to find an optimal schedule. We begin by constructing mathematical models to formulate the problem requirements, and then demonstrate the benefits of a polyhedral study on a relaxation of the physician scheduling problem that includes equalization constraints. A branch-and-cut algorithm using valid inequalities derived from the relaxation problem shows that the quality of the schedules with respect to the soft constraints is notably better. An example problem from a hospitalist department is discussed in detail, and improvements for other schedules representing different specialties are also presented.